Categorification of the Jones–Wenzl projectors

Abstract

The Jones–Wenzl projectors $p_n$ play a central role in quantum topology, underlying the construction of SU(2) topological quantum field theories and quantum spin networks. We construct chain complexes $P_n$, whose graded Euler characteristic is the “classical” projector $p_n$ in the Temperley–Lieb algebra. We show that the $P_n$ are idempotents and uniquely defined up to homotopy. Our results fit within the general framework of Khovanov’s categorification of the Jones polynomial. Consequences of our construction include families of knot invariants corresponding to higher representations of ${\mathrm{U}}_q\mathfrak{sl}(2)$ and a categorification of quantum spin networks. We introduce 6j-symbols in this context.